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Three-Dimensional Virial Theorem (Quantum Mechanics)

  1. Apr 8, 2007 #1
    1. The problem statement, all variables and given/known data
    (a) Prove the three-dimensional virial theorem:

    [tex] 2<t> = <r\cdot \nabla V> [/tex]

    (for stationary states)

    2. Relevant equations

    Eq. 3.71 (not sure if this applies to 3 dimensions, but I think so)

    \frac{d}{dt}<Q> = \frac{i}{\hbar}<[\hat H, \hat Q]> + \left<\frac{\partial \hat Q}{\partial t}\left> [/tex]

    where the last term is the explicit time dependance of the operator Q.

    3. The attempt at a solution

    Letting [itex] Q = \vec r \cdot \vec p [/itex]

    [tex] \frac{\partial \hat Q}{\partial t} = 0 [/tex]

    and for stationary states:

    [tex] \frac{d}{dt}<Q> = 0 [/tex]


    [tex] 0 = \frac{i}{\hbar}<[\hat H, \hat Q]> = \frac{i}{\hbar}<[T+V, \vec r \cdot \vec p]> [/tex]

    [tex] = \frac{i}{\hbar}(<T(\vec r \cdot \vec p)>-<\vec r \cdot \vec p T> + <V(\vec r \cdot \vec p)> - <\vec r \cdot \vec p V>) [/tex]


    [tex] <\vec r \cdot \vec p V> = <\vec r \cdot (\vec pV)> + <V(\vec r \cdot \vec p)> [/tex]


    [tex] 0 = \frac{i}{\hbar}(<T(\vec r \cdot \vec p)> - <\vec r \cdot \vec p T> - <\vec r \cdot (\vec p V)>) [/tex]
    Last edited: Apr 8, 2007
  2. jcsd
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